Approximation of the Diffusive Transport Terms

In order to study the principles of spatial discretization, we consider the steady state diffusion of a property iHn a one dimensional domain as sketched in Fig. 12.3. 

Considering the ID system in x-direction, the west side face of the grid volume is referred to by w and the east side grid volume face by e. The distances between the nodes W and P, and between nodes P and E, are identified by 6xwp and 5xpe, respectively. Similarly, the distances between face w and point P and between P and face e are denoted by Sx p and dxpe, respectively. 

The key step of the finite volume method is the integration of the governing equation over the grid cell volume to form a discretized equation at the nodal point P. By use of the Gauss theorem and the midpoint quadrature formula, the result is  

In order to achieve useful forms of the discretized equation, the interface diffusion coefficient P and the property gradient di /dx at the east e and west w faces are required. Linear approximations are frequently used to calculate the interface values and the gradients. The linear profile approximation of the property gradient is second order and widely used for the evaluation of the diffusive fluxes in (12.77). For the diffusive flux at position e, the approximation is then written as  

The parameter value of at the surface position e is determined by linear interpolation between points P and E. For uniform grids, and when the interface e is midway between the grid node points, is approximated as the arithmetic mean of and r, , given by F e (/ ,/ -I- F- e)I2. 

---